Any modern optical communication system employs channel coding in the form of forward error correction (FEC) to achieve a desired target bit error rate (BER) despite transmission impairments. Systematic FEC schemes insert redundancy, typically properly chosen parity checks, at the transmitter, and exploit this redundancy at the receiver to recover the transmit message. The use of FEC is associated with an increase of the gross data rate and guarantees an extended reach to the receiver or a next 3R regenerator.
Current 100G systems rely upon quaternary phase-shift keying (QPSK) and polarization division multiplexing (PDM) and reserve the parity checks on an overhead of typically between 6.69% and 25%. The baud rate is increased proportionally to the FEC overhead to accommodate the augmented gross data rate in the frequency spectrum.
In a wavelength division multiplexing (WDM) system, several channels are multiplexed in frequency over the same optic fiber at regular frequency spacing. Dense WDM (DWDM) systems are typically based on a 50 GHz grid. Modern networks offer the possibility to route individually each channel at the photonic layer by means of reconfigurable optical add-drop multiplexers (ROADMs), which provide per-channel selection and switching capability. This offers a promising approach to improve the spectral efficiency of a DWDM system by channel bundling. If n 50 GHz filters with 2·n flanks are replaced in each ROADM by a single n·50 GHz filter (with only 2 flanks over the same bandwidth), a net saving of the spectral bandwidth corresponding to 2·(n−1) flanks can be achieved. The resulting n-fold aggregated frequency slot can be used to transmit a bundle of channels that are routed together and are therefore logically regarded as a single super-channel. Whereas for a 50 GHz grid a FEC overhead of about 20% proved to be a reasonable compromise between FEC performance and filter penalty, the opportunity to pack several channels in a super-channel opens up again the question about the optimal gross data rate. Note that for a conventional fixed 50 GHz grid, a reduction of the baud rate and, thus, of the occupied spectrum cannot be exploited directly but rather leads to unused spectrum within each frequency slot. However, in case of channel bundling a lower FEC overhead allows packing more channels together, thus increasing the overall throughput per fiber. Therefore, channel bundling is a strong motivation to reduce the FEC overhead from the customary 20% to e.g. below 10% or even below 7%. Unfortunately, however, with a conventional coding approach, the lower overhead results also in a weaker signal-to-noise ratio (SNR) performance and a significant reach penalty.
Nowadays, most 100G PDM-QPSK commercial systems rely upon so-called differential transmission, meaning that the information is encoded in the phase difference between each symbol and its predecessor. One technology that could promise a significant reach improvement is non-differential transmission, which encodes the information in the absolute phase of the transmit signal. It can be seen that in the presence of strong FEC, the theoretical gain of a non-differential over differential transmission amounts to 1.0-1.5 dB in terms of signal-to-noise ratio (SNR). Unfortunately, however, without a proper reference, the transmit phase is known at the receiver only modulo the rotational symmetry of the signal constellation. For example, in the case of QPSK, the receiver is unable to tell whether the received constellation is in-phase with the transmit constellation or is rotated by a multiple of 90°.
In practice, the phase reference may be provided through the periodic insertion of so-called “pilot tones”, i.e. interspersed auxiliary symbols whose absolute phase is known to the receiver. In S. Zhang, X Li, P. Y. Kam, C. Yu and J Chen, “Pilot-assisted decision-aided maximum-likelihood phase estimation in coherent optical phase-modulated systems with nonlinear phase noise,” IEEE Photonics Technology Letters, vol. 22, no. 6, pp. 380-382, March 2010, pilot-assisted decision-aided maximum-likelihood phase estimation is introduced. In H. Zhang, Y. Cai, D. G. Foursa, and A. N. Pilipetskii, “Cycle slip mitigation in POLMUXQPSK modulation,” in Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference (OFC/NFOEC) 2011, Paper OMJ7, March 2011, the phase recovered from the pilot symbols is combined with the outcome of a standard M-th power algorithm to produce a pilot-assisted M-th power phase estimate. In M A. Castrillon, D. A. Morero and M. R. Hueda, “A new cycle slip compensation technique for ultra high speed coherent optical communications,” in IEEE Photonics Conference (IPC) 2012, pp. 175-176, September 2012, a pilot-assisted cycle slip detector and corrector processes the output of a conventional M-th power phase estimate. All these prior art solutions employ pilot symbols to improve the carrier phase recovery, but unfortunately, at the expense of the spectral efficiency and the rate loss due to a baud rate increase. Namely, since the optical channel is affected by a strong phase noise, the receiver must always track the carrier phase also between the pilot symbols, with a risk of generating an unwanted abrupt phase jump congruent with the rotational symmetry of the constellation. This jump, known as “cycle slip”, can be corrected only with the help of the next pilot tone and, consequently, produces an error burst whose length depends on the pilot symbol spacing. Obviously, frequent pilot symbols reduce the impact of cycle slips, but at the same time adversely affect the spectral efficiency and the baud rate of the system. So in summary, the achievable performance is constrained by the trade-off between resilience to cycle slips and pilot overhead.
One idea to diminish the baud rate without weakening the SNR performance is introduced in G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 55-67, January 1982. In this work, it is suggested to allocate the FEC redundancy in the signal space rather than in the frequency spectrum: by using a larger constellation than what is strictly necessary to support the desired bit rate, one provides room for additional FEC redundancy and, possibly, even potential for reducing the baud rate and shrinking the occupied bandwidth. This approach is referred to in the following as “constellation expansion”. While Ungerboeck only suggested constellation expansion for the electrical domain, its application to fiber-optic communications has been fostered in B. P. Smith and F. R. Kschischang, “Future prospects for FEC in Fiber-Optic communications,” IEEE Journal Select. Topics Quantum Electr., vol. 16, no. 5, September 2010, but has not yet found its way into commercial systems.
A particular application of the “constellation expansion” concept is described in S. L. Howard and C. Schlegel, “Differential turbo-coded modulation with APP channel estimation,” IEEE Trans. Comm., vol. 54, no. 8, August 2006; and in S. Howard, “Differentially-encoded turbo-coded modulation with APP phase and timing estimation,” Ph.D. thesis, University of Alberta, Canada, November 2006. In this work, it is shown that a concatenation of a single parity check (SPC) code with a differential 8-ary phase-shift keying (8PSK) modulation may outperform even large 8PSK trellis codes as suggested by Ungerboeck by 1 dB. The simplicity of the SPC code makes this solution attractive also for optical communications, where the extremely high data rate sets a limit to the complexity of viable algorithms. Howard and Schlegel propose a combination of decoder and carrier recovery according to an iterative expectation-maximization (EM) algorithm: at every iteration an expectation step calculates a soft-decision phase reference on the basis of the current decoding result, and a maximization step uses this phase reference to compute the channel metrics for the next decoder run. Unfortunately, however, the resulting carrier recovery fails to fulfil the requirements posed by the rough fiber-optic communication channel.
A first problem encountered is the existence of meta-stable states: For specific critical channel phases, the acquisition algorithm can hang indefinitely in an invalid state even if a very large number of decoding iterations is performed. This situation results, of course, in a dramatic degradation of the BER. Further, although the carrier recovery does not entail an M-th power computation, the expectation-maximization algorithm can generate cycle slips due to the rotational invariance of the differential 8PSK modulation. At each cycle slip bit errors occur, which, depending on their distribution with respect to the SPC code, are corrected or persist during the subsequent decoding process. The cycle slips are also the reason of a limited tolerance of this scheme to frequency offset. A frequency offset produces continuous cycle slips that overload the decoder.
Finally, the expectation-maximization algorithm employed by Howard and Schlegel is not sufficiently robust against phase noise. Laser oscillators have Lorentzian line-shape, which gives rise to a Wiener phase noise process, i.e. a Gaussian phase random walk (see e.g. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” IEEE Journal Lightwave Technology, vol. 22, no. 7, April 2009). With the current technology, the linewidth of the transmit laser and the local oscillator laser at the receiver may lie between 100 kHz and 1 MHz. For a PDM-QPSK 100G system and a realistic combined laser linewidth in the range of 200 kHz to 2 MHz, the standard deviation of the phase steps generated by laser phase noise ranges from 0.38 to 1.2 degrees per symbol interval. Nonlinear effects along the fiber may further increase the phase noise. In section 9.6.2 of the above-cited PhD-thesis of S. Howard, results in the presence of a random walk phase process are presented. It is seen that with a standard deviation of only 0.22 degrees per symbol interval, a penalty as high as 0.25-0.5 dB is observed along the so-called “turbo cliff”.
In the literature, more advanced techniques for joint carrier recovery and decoding have been proposed. An overview is given in G. Colavolpe and G. Caire, “Iterative joint detection and decoding for communications under random time-varying carrier phase,” Tech. Rep., ESA Contract 17337/03/NL/LvH, March 2004, which shows the superiority of the Bayesian approach over expectation-maximization algorithms. The algorithms described in G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J Sel. Areas Commun., vol. 23, no. 9, pp. 1748-1757, September 2005 for non-differential transmission, and in A. Barbieri and G. Colavolpe, “Soft-output detection of differentially encoded M-PSK over channels with phase noise,” EUSIPCO 2006, Florence, Italy, September 2006 for differential transmission achieve almost optimal performance also in the presence of very strong phase noise and, at the same time, are simpler than most alternative solutions. Unfortunately, however, their complexity remains too high for an implementation at optical data rates.